reserve i,j for Nat;

theorem Th9:
  for n,m being Nat,K being Ring holds -(0.(K,n,m))=0.( K,n,m)
proof
  let n,m be Nat,K be Ring;
  per cases by NAT_1:3;
  suppose
A1: n>0;
A2: 0.(K,n,m) + 0.(K,n,m) = 0.(K,n,m) by MATRIX_3:4;
A3: len (0.(K,n,m))=n by MATRIX_0:def 2;
    then width (0.(K,n,m))=m by A1,MATRIX_0:20;
    hence thesis by A3,A2,Th8;
  end;
  suppose
    n = 0;
    then
A4: len (0.(K,n,m)) = 0;
    then len (-(0.(K,n,m))) = 0 by MATRIX_3:def 2;
    hence thesis by A4,CARD_2:64;
  end;
end;
